Extremal storage functions and minimal realizations of discrete-time linear switching systems

We study the L
_p
induced gain of discrete-time linear switching systems with graph-constrained switching sequences. We first prove that, for stable systems in a minimal realization, for every p ≥ 1, the L
_p
-gain is exactly characterized through switching storage functions. These functions are shown to be the pth power of a norm. In order to consider general systems, we provide an algorithm for computing minimal realizations. These realizations are rectangular systems, with a state dimension that varies according to the mode of the system. We apply our tools to the study on the of L
_p
-gain. We provide algorithms for its approximation, and provide a converse result for the existence of quadratic switching storage functions. We finally illustrate the results with a physically motivated example.